Question

If alpha and beta are the zeroes of the quadratic polynomial f(x)=x^2-2x+3, find a polynomial whose roots are alpha-1/alpha+1, beta-1/beta+1

Answer 1

Final Answer:
$3 {x}^{2} - 2x + 1 = 0$
Steps:
1) Let a, b be zeroes of the quadratic polynomial
${x}^{2} - 2x + 3 = 0$

2) Then, quadratic polynomial whose zeroes are
$\frac{(a - 1)}{(a + 1)} = x$
By componendo and dividendo,
we get
$a = \frac{(x + 1)}{(1 - x)}$
Therefore,a satisfies the given equation.
${ (\frac{(1 + x)}{(1 - x)} )}^{2} - 2 \frac{(1 + x)}{(1 - x)} + 3 = 0 \\ = > {( x+ 1)}^{2} + 2( {x}^{2} - 1) + \\ 3 {(1 - x)}^{2} = 0 \\ = > 3 {x}^{2} - 2x + 1 = 0$

Therefore, Quadratic polynomial whose zeroes are
$\frac{(a - 1)}{(a + 1)} \: \: \: and \: \frac{(b - 1)}{(b + 1)}$
is
$3 {x}^{2} - 2x + 1 = 0$